Using tensor product splines in modeling exposure-time-response relationships: application to the Colorado Plateau Uranium Miners cohort

Abstract

An adequate depiction of exposure-time-response relationships is important in assessing public health implications of an occupational or environmental exposure. Recent advances have focused on flexible modeling of the overall shape of latency. Methods are needed to allow for varying shapes of latency under different exposure profiles. A tensor product spline model is proposed for describing exposure-response relationships for protracted time-dependent occupational exposure histories in epidemiologic studies. The methods use flexible multi-dimensional techniques to jointly model age, latency and exposure-response effects. In analyzing data from the Colorado Plateau Uranium Miners cohort, a model that allows for varying exposure-dependent latency shapes is found to be superior to models that only allowed for an overall latency curve. Specifically, the model suggests that, at low exposure levels risk increased at short latencies followed by a slow decline for longer latency periods. On the other hand, risk was higher but did not change much by latency for higher exposure levels. The proposed methodology has the advantage of allowing for latency functions that vary by exposure levels and, conversely, exposure-response relationships that are influenced by the latency structure.

Figure 1

Relative lung cancer risk for Colorado Plateau Uranium Miners as a function of latency (in years) and annual radon exposure levels (in WL of radon exposure). The two-dimensional tensor product spline-based model used a no-knot cubic spline (i.e. a cubic polynomial) for latency and a cubic spline with an interior knot at the median value of 63 WLs for radon exposure.

Figure 2

Cross sections of the relative lung cancer risk surface (based on the surface in Figure 1) for Colorado Plateau Uranium Miners as a function of latency (in years) and annual radon exposure levels (in WL of radon exposure). Panel (a) gives exposure–response curves for various levels of latency (in years) and panel (b) depicts the latency curves for various levels of exposures (in WLs).

Figure 3

Relative risk estimates as functions of latency for ‘low,’ ‘medium’ and ‘high’ categories of exposure given by <42, [42,92] and >92 WLs of radon exposure, respectively.